CARDSTOCK MODELING

Math Manipulative Kit

Revised July 25, 2006

TABLE OF CONTENTS

Unit Overview..........................................................................................................3 Format & Background Information ..........................................................................3-5 Learning Experience #1 - Getting Started ...............................................................6-7 Learning Experience #2 - Squares and Cubes (Hexahedrons) ..............................8-10 Learning Experience #3 - Triangles and Tetrahedrons ...........................................11-13 Learning Experience #4 - Volume and Surface Area ..............................................14-18 Learning Experience #5 - Platonic Forms ...............................................................19-20 Learning Experience #6 - Comparing Interior and Exterior Angles .........................21-22 Learning Experience #7 - Proving Formulas ...........................................................23-26 Learning Experience #8 - Modeling the Planets......................................................27-30 Glossary ..................................................................................................................31-34

Unit Overview

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Three dimensional models provide a basis for many interesting math, science, and design experiences. This card modeling unit provides a unique way for students to explore a variety of geometric concepts that reach out from the traditional two dimensional models typically used in instruction. This flexible system allows students to explore area, perimeter, and the properties and concepts of solid geometric shapes. It provides an opportunity to scale and compare the various sizes of planets within our solar system, and to make proportional comparisons of the distances between the planets in the solar system and the distances of objects here on Earth. Scheduling This unit may take from a day if you chose to use only one of the card modeling activities to the entire year to complete depending upon the goals of the teacher and interests of the students. Materials to be obtained locally: Please make one student activity book for each student. pencils scissors square template cube form triangle template posterboard white paper tetrahedron form pentagon template glue water About the Format Each learning experience is numbered and titled. Under each title is the objective for the learning experience. Each learning experience page has two columns. The column on the left side of the page lists materials, preparations, basic skill processes, evaluation strategy, and vocabulary. The evaluation strategy is for the teacher to use when judging the student’s understanding of the learning experience. The right column begins with a “Focus Question” which is typed in italicized print. The purpose of the “Focus Question” is to guide the teacher’s instruction toward the main idea of the learning experience. The “Focus Question” is not to be answered by the students. The learning experience includes direction for students, illustrations, and discussion questions. These discussion questions can be used as a basis for class interaction.

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Background Information Creating Polyhedra shapes A three-dimensional shape that has many flat surfaces is called a polyhedron. The “hedron” part of the word polyhedron is of Greek origin and means “flat surface”. The prefix “poly” means “many”. So the word “polyhedron” means a three-dimensional shape with many flat surfaces. A flat surface of a solid figure is called a face. The line where two faces meet is called an edge. The point where several edges meet is called a vertex. The five regular polyhedrons are known as Platonic solids named after the Greek philosopher Plato. He wrote about them in his dialogue “Timaeus”. The polyhedrons represent the physical elements of the world. Platonic solids:

• Look the same when viewed at any corner, edge, or center of any face. • Have congruent regular polygon faces. • Have congruent edges, face angles, and corners.

A regular polyhedron or Platonic solid is a polyhedron with the following properties:

a. All faces are regular polygons. b. All faces are congruent to each other. c. The same number of faces meets at each vertex in exactly the same way.

A polyhedron is a three-dimensional shape formed by joining edges of polygons to enclose a region of space. The polygons are called the faces of the polyhedron. Exactly two polygons meet at each edge of the polyhedron. At least three faces meet at each vertex of the polyhedron.

Number of faces Polyhedron Shape of polygon face

Number of faces at each vertex

6 Cube/Cuboid Squares 3 4 Tetrahedron Triangles 3 8 Octahedron Triangles 4

20 Icosahedron Triangles 5 12 Dodecahedron Pentagons 3

In a regular polygon, all sides have the same length and all angles have the same number of degrees of measure. Therefore it is both equilateral and equiangular. Two polygons are congruent if their corresponding sides are equal in length and their corresponding angles are equal in measure. Two congruent polygons have the same size and shape.

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Meanings of prefixes Tri- = 3 Penta- = 5 Hexa- = 6 Octa- = 8 Quad- = 4 Do (as in double) = 2 Deca ( as in decade) = 10 Dodeca = 2 + 10 or 12 -gon = angle Polyhedral shapes are found everywhere in the modern world of complex structures. For example: building, packages, boxes, rooms, doors, cabinets, etc. Polyhedral shapes are not as common in nature. However when the crystal make-up of materials such as rocks, metals, and powders are viewed through microscopes, they have polyhedral shapes. Scientists who study the molecular structure of such materials must have detailed knowledge of polyhedral shapes to understand the structure of matter. Perimeter Perimeter is the linear distance around an object or figure. Also the boundary of a closed plane figure. Area Area is the surface included within a set of lines; specifically: the number of square units (e.g., square inches, square centimeters) a figure contains. The square units used to measure area are often based on multiplying units of length and width. Volume Volume is the amount of space occupied by a three-dimensional object as measured in cubic units (e.g., cubic inches, cubic centimeters, liters). The cubic units used to measure volume are often based on multiplying units of length, width, and height. Cardstock Modeling The geometric panels used in this kit easily attach along the edges of panels with rubberbands to create various polyhedra. Students use these hands-on manipulative to learn the names and properties of polygons such as triangle, rectangle, square, parallelogram, pentagon, hexagon, and octagon. Assembly of solid shapes with panels leads to exploration of basic properties of the five regular polyhedra (tetrahedron, cube, octahedron, icosahedrons, and dodecahedron) and pyramids. The method of connecting flat panels with rubberbands was invented by Fred Bassetti, an architect who was interested in rapid assembly of polyhedral shapes to aid in design of new structures.

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Learning Experience 1: Getting Started Objective: Students will review the vocabulary necessary to complete this unit through the cooperative learning strategy known as “guess the fib”.

To begin heterogeneously group your students by skill level into five groups. Each groups receives a partial list of the vocabulary to be reviewed, 11 words total, and the students in each group will divide up those 11 words between them. Each student should have all 11 words written down when they are done. Each student is to write two facts and one “believable fib” about each of their words on the sentence sheet provided in Learning Experience #1 in the Cardstock Modeling Student Activity Book. If they have difficulty with certain words, allow them to use the Glossary in the back the their student activity book. When all groups are ready, read each word individually. Have a student from the group that has that word announce all three statements. Be sure to have them mix them up as they read them or students will start to see that the fib is always at the end. It is the job of the other groups to guess which statement is the fib. They can do this by writing down the false statement and then state what they believe is false after all groups have recorded their false statements. This goes on until all 55 words have been reviewed. Keep score to determine which group was able to fool the most teams at the end of the game.

Materials: For each group of students: Cardstock Modeling Student Activity Book Vocabulary sheet for Learning Experience #1 cut into group lists Glossary of terms (if needed) Preparation: Copy the team vocabulary lists and cut them into strips by group. Distribute a list to each group. Basic Skills Development: Recall of material differentiation of truth over fiction. Evaluation Strategy: Monitor student response. Vocabulary: area base centimeter concentric congruent corner cube customary diameter dodecahedron edge equilangular equilateral equilateral triangle exterior angle face form hexagon hexahedron icosahedron interior angle interior line segment intersection lateral line segment model octahedron panel parallel parallelogram pentagon perimeter perpendicular platonic solidspoint polygon polyhedron prism proportional pyramid quadrilateral radius rectangle regular polygon right angle scientific notation square surface area template tetrahedron trapezoid triangle vertex vertices volume

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Learning Experience 1 continued Page 2 Vocabulary List for Team 1: area base centimeter concentric congruent corner cube customary diameter dodecahedron edge Vocabulary List for Team 2: equilangular equilateral equilateral triangle exterior angle face form hexagon hexahedron icosahedron interior angle interior line segment Vocabulary List for Team 3: intersection lateral line segment model octahedron panel parallel parallelogram pentagon perimeter perpendicular Vocabulary List for Team 4: platonic solids point polygon polyhedron prism proportional pyramid quadrilateral radius rectangle regular polygon Vocabulary List for Team 5: right angle scientific notation square surface area template tetrahedron trapezoid triangle vertex vertices volume

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Learning Experience 2: Squares and Cubes (Hexahedrons)

Objective: Students will create square cardboard panels and measure their sides and angles. Students will use the square panels to form a cube and identify/count its faces, corners, edges, and faces at each vertex.

Session 1: Student pairs will engage in the use of measurement tools – metric/customary rulers and protractors to analyze the characteristics of the drawing provided in Learning Experience #2 in the Cardstock Modeling Student Activity Book. Session 2: In this learning experience, students will create a cube or hexahedron in its three-dimensional form using panels made from lightweight card paper. Rubberbands will be used as connectors. To begin, create templates of the square shape. The number of templates created is dependent upon the number of students in your classroom. Templates can be shared among students. The pattern of the square templates can be found on page 9 of the Teachers Guide. Directions for creating template: 1. Attach the pattern to a piece of

cardstock with a glue stick. 2. Punch a hole at the location of the

center intersection of the interior lines on the pattern with a 1/8” hole punch.

Panel Template Pattern

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Materials: For each pair of students: 2 Cardstock Modeling Student Activity Books Cardstock Metric ruler #14 rubberbands Protractor Pencil* Scissors* Square template*

For the class: 5 glue sticks 5 1/8 “hole punch (for template) 5 ¼ “hole punch (for panels)

*provided by teacher Preparation: You must make copies of the student manual. Note: Check to insure that the copies are made with the correct degree of enlargement or reduction. The Xerox copies should measure 8 cm on the interior edge. Create templates for students and label them TEMPLATE. The templates will be used to create the panels. Create models of the cube for student reference. Note: Verify that each line segment measures 8 cm.

Basic Skills Development: Observing Comparing Measuring Manipulating Materials Discussing

Evaluation Strategy: Students will create square panels to form a cube and use various measurement tools to identify the size/number of edges, angles, faces, and corners.

Vocabulary: template interior angle perimeter panel exterior angle customary polygon perpendicular right angle square cube corner centimeter hexahedron edge face vertex parallel point vertices intersection line segment regular polygon interior line segment

Learning Experience 2 continued Page 2 3. Cut out the shape along the exterior lines. The templa(hexahedrothe vertex folded on tline will be Before cretheir templModeling S

Students afrom the te

tes may now be used by students to develop the panels that create the cube n). The interior line segment of the template must measure 8 cm between

of each intersection. The panels created from the template will eventually be he dotted 8 cm line segment. An additional line that is 8 mm from the fold come the exterior edge of the panel.

ating the panels for the cube, ask students to answer questions 1-10 about ate on the activity sheet for Learning Experience #2 in the Cardstock tudent Activity Book.

re to follow the directions on page 3 of the activity sheet to create panels mplate. Students are to create six (6) square panels for the cube assembly.

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Learning Experience 2 continued Page 3 Once student panels are created, the teacher should demonstrate how to create a model of a cube using rubberbands. The model is constructed by placing rubberbands over the notches in the end of each panel. As more panels are added, the model will take on a three-dimensional form. Guide students through completing the chart on page 4 of the activity sheet for Learning Experience #2.

Form Polygon(s) used for

faces

Number of sides of

each face

Number of faces

Number of corners

Number of edges

Number of faces at

each vertex

Cube (hexahedron)

square 4 6 8 12 3

Answers to activity sheet: Metric Customary 1. 9.6 cm 3 13/16” 2. 8 cm 3 1/8” 3. 8 mm 5/16” 4. 32 cm 12 ½ “ 5. 90° 90° 6. 90° 90° 7. 360° 360° 8. 360° 360° 9. Yes, the angles are the same and the sides are proportional. 10. Yes, all sides have the same length and all angles have the same number of degrees of measure.

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Learning Experience 3: Triangles and Tetrahedrons Objective: Students will create triangle cardboard panels and measure the

sides and angles. Students will use the triangle panels to form a tetrahedron and identify/count its faces, corners, edges, and faces at each vertex.

Session 1: Students will engage in the use of measurement tools – metric/customary rulers and protractors to analyze the characteristics of the drawing provided in Learning Experience #3 in the Cardstock Modeling Student Activity Book.

Session 2:

Student pairs will create a second three-dimensional form from cardstock paper and rubberbands used as connectors. To begin create templates of the triangle shape. The number of templates created is dependent upon the number of students in your classroom. Templates can be shared among students. The patterns for the triangle template can be found on page 12 of the Teacher’s Guide. Directions for creating the template: 1. Attach the pattern to a piece of

cardstock with a glue stick. 2. Punch a hole at the location of the

center intersection of interior lines on the pattern with a 1/8” hole punch

Panel Template Pattern

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Materials: For each pair of students: 2 Cardstock Modeling Student Activity Books Cardstock Metric ruler #14 rubberbands Protractor Pencil* Scissors* Triangle template*

For the class: 5 glue sticks 5 1/8”hole punch (for template) 5 ¼ “hole punch (for panels)

*provided by teacher Preparation: You must make copies of the student manual. Note: Check to insure that the copies are made with the correct degree of enlargement or reduction. The Xerocopies should measure 8 cm on the interior edge. Create templates for students. The templates will be usedcreate the triangle panels. Create modof the tetrahedron for student refere

x

to els

nce. Basic Skills Development: Observing Comparing Discussing Measuring Manipulating Materials

Evaluation Strategy: Students will create triangle panels to form a tetrahedron and use various measurement tools to identify the size/number of edges, angles, faces, and corners.

Vocabulary: template polygon congruent panel perimeter line segmenttriangle interior angle parallel centimeter exterior angle edge tetrahedron corner face point perpendicular equilateral triangle

Learning Experience 3 continued Page 2 3. Cut out the shape along the exterior lines.

The templates matetrahedron. The The panels createadditional line tha Before creating thabout their templaModeling Student

Students are to thpanels from the tecreation of a tetra

Once student panusing rubberband

y now be used by students to develop the panels that create a template will measure 8 cm between the vertex of each intersection. d from the template will eventually be folded on the 8 cm line. An t is 8 mm from the fold line will become the exterior edge of the panel.

e panels for the tetrahedron, ask students to answer questions 1-10 te on the activity sheet for Learning Experience #3 in the Cardstock Activity Book.

en follow the directions on page 2 of the activity sheet to create mplate. Students are to create four (4) triangle panels for their hedron assembly.

els are created, demonstrate how to create a model of a tetrahedron s. The model is constructed by placing rubberbands over the notches

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Learning Experience 3 continued Page 3 in the end of each panel. As more panels are added, the model will take on a three-dimensional form. Guide students through completing the chart on page 3 of the activity sheet for Learning Experience #3.

Form Polygon(s) used for faces

Number of sides of

each face

Number of faces

Number of corners

Number of edges

Number of faces at

each vertex

Tetrahedron triangle 4 4 5 6

3

Answers to activity sheet:

Metric Customary 1. 10.9

cm 3 13/16”

2. 8 cm 3 1/8” 3. 8 mm 5/16” 4. 24 cm 9 3/8” 5. 60° 60° 6. 120° 120° 7. 180° 180° 8. 360° 360°

9. Equilateral triangle. 10. The triangle has three congruent sides.

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Learning Experience 4: Volume and Surface Area Objective: Students will measure the sides of the cube, tetrahedron, and pyramid and use the appropriate formulas to find their volume and/or surface area.

Session 1: In this learning experience, student pairs will use appropriate formulas to find the volume of the cube they created with the panels and create a pyramid with the panels and find its volume. To find the volume of the cube, students are to use the formula:

Cube = a3

Students are to show their work on the activity sheet for Learning Experience #4 in the Cardstock Modeling Student Activity Book. Cube = a 3 = 8 3

= 512 cm3

Check student calculations of volume by using the 500 mL graduated cylinder and measure the volume of the cube using beads. First show students the relationship between cm3 and mL

a

a

a

Materials: For each pair of students: 2 Cardstock Modeling Student Activity Books Cardstock Metric ruler #14 rubberbands Container of Beads Pencil* Scissors* Cube form* Tetrahedron form* Triangle template* For the class: 5 glue sticks 5 ¼” hole punch 25 mL graduated cylinder 500 mL graduated cylinder Centimeter cube Funnel Water* *provided by teacher Preparation: Create model of the pyramid for student reference. Basic Skills Development: Comparing Measuring Discussing Manipulating Materials Evaluation Strategy: Students will use appropriate formulas to find the volume and/or surface area of cube, pyramid, and tetrahedron. Vocabulary: cube tetrahedron pyramid volume surface area area base form difference square triangle

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Learning Experience 4 continued Page 2 to illustrate how we can calculate the volume in cm (cm3) and show relative volume using mL. Fill the 25 mL graduated cylinder with 10 mL of water. Find the volume of the centimeter cube using linear measurement (length x width x height or a3). A funnel has been included to aide with pouring of materials.

a3= 1 cm x 1 cm x 1 cm = 1 cm 3

Drop the cube into the graduated cylinder and ask students to observe how much higher the water is displaced with the centimeter cube. The water is displaced 1 mL. Therefore: 1 cm3 = 1 mL. Using the large graduated cylinder measure the volume of the cube in mL with beads. Lift the top panel of the cube and pour the beads into the cube. It should fill the cube displaying its volume. To find the volume of a pyramid, ask students to create another square panel and remove the bottom panel from the tetrahedron using that to create another lateral face. The square will then become the base of the pyramid. Students are to use the formula:

Volume = 1/3 Bh,

where B represents the area of the base. area of the base is multiplied by the height divided by three (3). Pyramid = 1/3 Bh = 1/3 (8 x 8) 5.65 = 1/3 (64) 5.65 = 1/3 (361.6) cm3

= 120.53 cm3 Again show relative volume with

1 cm

1 cm

1 cm

To calculate the height of the triangle: 6.928 = slanted side 42 + x2 = (6.928)2

2

6.928

x

beads to check calculations.

16 + x ≈ 48 x ≈ √32

x ≈ 5.65 4

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Learning Experience 4 continued Page 3 Notice the height of the pyramid does not equal 8 cm. To compare the volume of the pyramid as 1/3 the size of the cube, the height of the pyramid must equal 8. To create this size pyramid, the panels for our pyramid must change. They will no longer be equilateral triangles. Use Pythagorean theorem to find the size of the panels for the new pyramid.

9.79

8.94

4 cm

4 cm

8 cm

Size of panels for pyramid 8 cm in height.

42 + 82 = x2

16 + 64 = x2

80 = x2

√80 = x 8.94 ≈ x

4 cm

x Center height in pyramid.

8 cm

42 + 8.942 = x2

16 + 79.9 = x2

95.9 = x2x

4 cm

Students can then create four (4) panels ofUse the volume formula to find volume: If you multiply 170.66 by 3, it should be appsince the volume of the pyramid is 1/3 that must be of same height to make this compa

8.94 cm

√95.9 = x 9.79 ≈ x

size shown in box above for new pyramid.

Volume = 1/3 Bh = 1/3 (8 x 8) 8 = 1/3 (64) 8

= 1/3 (512) ≈ 170.66

roximately equal to the volume of the cube of the cube. However the cube and pyramid rison.

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Learning Experience 4 continued Page 4 Session 2: Students are to complete the chart on page 2 of the activity sheet for Learning Experience #4 in the Cardstock Modeling Student Activity Book to find the surface area of the cube, tetrahedron, and pyramid.

Platonic form

Shape of base

Area of base (B)

Other Surfaces

Area of Other

Surfaces

Surface Area

Cube Square B = wl = (8) (8) = 64 cm2

5 Squares (64) (5) = 320 cm2

384 cm2

Tetrahedron Triangle B = ½ wl = ½ (8) (7) = ½ (56) = 28 cm2

3 Triangles (28) (3) = 84 cm2

112 cm2

Pyramid Square B = wl = (8) (8) = 64 cm2

4 Triangles B = ½ wl = ½ (8) (7) = ½ (56) = 28 cm2

(28) (4) = 112 cm2

176 cm2

Note: The height of the tetrahedron is √48 ≈ 7

+

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Learning Experience 4 continued Page 5 Students also find the difference between the surface area between the tetrahedron and pyramid. They will find the difference to be 36 cm2. This difference is due to the difference in the area between the base of the tetrahedron (a triangle) and the base of the pyramid (a square). Area of base of tetrahedron B = ½ wl = ½ (8) (7) = ½ (56) = 28 cm2 Area of base of pyramid

Difference:

64 cm2 – 28 cm2 = 36 cm2

Note: The height of the tetrahedron is √48 ≈ 7

B = wl = (8) (8) = 64 cm2 Answers to activity sheet: 1. a3 = 8x8x8 = 512 cm3

2. 1/3 Bh = 1/3 (8 x 8)7 = 1/3 (64)7 = 1/3 (448) cm3 = 149 1/3 cm3 3. See chart on page 12 of Teacher’s Guide. 4. See explanation above.

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Learning Experience 5: Platonic Forms

Objective: Students will create panels to form an octahedron, dodecahedron, and icosahedron and identify/count its faces, corners, edges, and faces at each vertex and use a formula to find the number of vertices and edges.

In this learning experience, students will be creating three-dimensional models of the remaining Platonic forms: octahedron, dodecahedron, and icosahedron. Divide the class into three groups. Each group will create panels for a specific form pictured on page 1 of the activity sheet for Learning Experience #5 in the Cardstock Modeling Student Activity Book. Students should also complete the chart for each form on page 1 of the activity sheet. Student groups can then share their results with other groups that created the same form and the rest of the class. Point out to students that while it may be easy to count the faces, edges, and vertices of some Platonic solids, it is not so easy for more complicated polyhedrons. Page 2 of the activity sheet for this learning experience leads students through the “shortcut” to finding the number of vertices and number of edges of these more complicated polyhedrons. The explanation of each step is described below: To find the number of edges:

Form # of faces

# of edges

on each face

# faces that

share edge

Dodecahedron 12 5 2 ( 12 x 5 ) / 2 = 30

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Materials: For each group of three of students: 3 Cardstock Modeling Student Activity Books Cardstock Metric ruler #14 rubberbands Pencil* Scissors* For the class: 5 ¼” hole punch 5 1/8” hole punch 5 glue sticks *provided by teacher Preparation: Create pentagon templates for students. The templates will be used to create the panels for the dodecahedron. Create models of the various platonic forms for student reference. Basic Skills Development: Manipulating Materials Discussing Measuring Comparing Evaluation Strategy: Students will create panels to form the platonic solids and use formulas to find the number of faces, corners, edges, and faces at each vertex. Vocabulary: template perimeter panel polygon triangle hexagon square rectangle centimeter pentagon point icosahedron face octahedron corner dodecahedron edge polyhedron parallel line segment # of

edges # of edges on each face

# of faces share edge

# of faces

Learning Experience 5 continued Page 2 In the calculation, (12x5) = # of faces x # of edges per face, each edge of polyhedron is counted twice since each edge is shared by 2 faces. Therefore, we must divide by 2. To find the number of vertices:

Form # of faces # of vertices on each face

# of faces that meet at each

vertex Dodecahedron 12 5 3 ( 12 x 5 ) / 3 = 20

# of vertices

# of faces meet at each vertex

# of vertices on each face

# of faces

In the calculation (12 x 5) = # of faces x # of vertices per face, each vertex of this polyhedron is counted 3 times, since each vertex is shared by 3 faces. Therefore, we must divide by 3. Students can explain the process of counting edges and vertices and describe shortcuts.

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Learning Experience 6: Comparing Interior and Exterior Angles

Objective: Students will compare the interior and exterior angles of five polygons and analyze data for patterns.

Review the meaning of interior and exterior angle of a polygon. Ask each group of students to create a panel of the pentagon, hexagon, and octagon from the templates provided. Students should not bend these panels as they will use them to measure the angles of the polygons. Students will then complete the activity sheet for Learning Experience #6 in the Cardstock Modeling Student Activity Book. 1. From the chart students will look for patterns in their data. Answers to activity sheet: Polygon # of

sides Interior angle

Sum of interior angles

Exterior angle

Sum of exterior angles

triangle 3 60 180 120 360 square 4 90 360 90 360 pentagon 5 108 540 72 360 hexagon 6 120 720 60 360 octagon 8 135 1080 45 360

2. Students should see that the sum of

the exterior angles of any regular polygon is 360°.

To demonstrate that the sum of the interior angles of any regular polygon is 180°, have the students cut any size triangle out of a piece of paper. Then have them shade in the tip of each corner and rip off the corners. Next, place the outside points of each corner together to form a straight edge.

Materials: For each pair of students: 2 Cardstock Modeling Student Activity Books Cardstock Metric ruler Protractor Pencil* Scissors* Pentagon template*

For the class: 5 1/8” hole punch 5 ¼” hole punch 5 glue sticks

*provided by teacher

Preparation: Create hexagon and octagon templates for student groups.

Basic Skills Development: Comparing Discussing Measuring Analyzing Data

Evaluation Strategy: Students will analyze measurement data of various polygons and find patterns within the data to make conclusions about these polygons.

Vocabulary: template panel triangle square pentagon hexagon octagon polygon interior angle exterior angle

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Learning Experience 6 continued Page 2 3. Other patterns:

• As the number of sides increases, the measure of each exterior angle decreases. • As the number of sides increases, the measure of interior angles increases. • The sum of interior angle and exterior angle is 180°. • Dividing 360° by the number of sides gives the exterior angle. • Subtracting the exterior angle from 180° gives the interior angle.

4. 360°/ 7 = 51.42 = 51° 5. 180° - 51= 129°

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Learning Experience 7: Proving Formulas Objective: Students will use panels of various shapes to prove the area formula for a square, triangle, parallelogram, and trapezoid.

Students use various formulas to find the area of various shapes. This learning experience has students manipulating various panels they have already worked with in previous learning experiences to explain/show why specific formulas are used to find the area of a square, the triangle, parallelogram, and trapezoid. Students are to follow the directions on the activity sheet for Learning Experience #7 in the Cardstock Modeling Student Activity Book to take them though this learning experience. Student pairs will need to cut out four (4) triangle panels and nine (9) square panels from a template before working on the activity sheet. In the first part of the activity sheet, students use the square panels to create larger squares proving the area formula for a square. Students are then finding the actual area of the various size squares in centimeters. Using the number sequence 1, 4, 9, 16, 25, 36, students see the number relationships in finding the area of the squares.

Materials: For each pair of students: 2 Cardstock Modeling Student Activity Books Metric rulers Cardstock Graph paper Pencil* Square template* Triangle template* Scissors* *provided by teacher Preparation: Create any additional templates necessary for students to create panels. Basic Skills Development: Manipulating Materials Comparing Discussing Observing Evaluation Strategy: Students will use various shaped panels to prove area formulas of a square, triangle, parallelogram, and trapezoid. Vocabulary: area triangle square polygon rectangle parallelogram trapezoid parallel quadrilateral

1

4

9

+3

+5

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Learning Experience 7 continued Page 2 Next students are to measure 23 cm x 19.5 cm rectangle on graph paper and cut it out. Then students are place four triangle panels on the rectangle to form a larger equilateral triangle. Students then trace the large equilateral triangle and cut it out. They will see that two (2) right triangles remain. Students should be able to fit the two right triangles on top of the equilateral triangles proving: The area of a rectangle = wl, however, the area of a triangle = ½ wl because in a rectangle there are two triangles. Therefore, you must divide wl in ½ to find the area of one triangle.

Then using two (2) triangle panels, students create a parallelogram that is then traced on graph paper. Students then cut the extended corner on the right side of the parallelogram and place it on the left side of the parallelogram. This forms a square, which proves why we can use the same formula to find the area of a square and a parallelogram. Students then are finding different ways to find the area of a trapezoid. Students form a trapezoid with three (3) triangle panels and trace the trapezoid on graph paper. The formula for a trapezoid is then broken down for students to put in measurements. It then asks students to check their work by finding the area of the three equilateral triangle panels that we used to create the trapezoid and add those areas together to find the area of the trapezoid. A second way students find the area of a trapezoid is to form two (2) triangles out of the trapezoid and find the area of the two (2) triangles and adding the areas together to find the areas of the trapezoid. Students then create a larger parallelogram with two (2) trapezoids. Students can find the area of the larger parallelogram by using the formula for area of parallelogram, by finding the area of the trapezoid and multiply it by 2, or by finding the area of the triangles used to create the parallelogram and multiply by 6. Students are to come up with these different ways to find the area of the parallelogram by looking at the shape in a variety of ways. Finally students complete the chart on page 7 of the activity sheet using the formula for the area of a rectangle. Algebraic expressions are also to be used by students with the area of a triangle and trapezoid formulas. Additional algebraic expressions could be created by student groups within the various shapes used in this learning experience and then students could complete area formula with these expressions.

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Learning Experience 7 continued Page 3 Answers to activity sheet: 1. 2. 19.2 cm x 19.2 cm3. 28.8 cm x 28.8 cm4. Using the numberrelationships in findinthree (3), gives you fcreate the next large5. 49, 64, 81, 100 6. When the two (w) equilateral triangle is7. The area of a rectrectangle there are twone triangle. 8. Lines in the same 9. A quadrilateral wit10. The extended coleft side of the paralleformula to find the ar11. 22 + 11 = 33

b1 b2 12. 33/2 = 16.5 13. 16.5 x 9.5 = 15614. ½ (11) (9.5) = 5215. Area of Triangle I16. Area of Triangle I17. Area of Trapezoid18. 33 x 9.5 = 313.5 19. Area of trapezoid20. Area of one triang

B=wl = 3 x 3 = 9 Nine square panels are used to create a larger square with three panels on each side.

= 368.64 cm2 = 369 cm2 = 829.44 cm2 = 829 cm2 sequence 1, 4, 9, 16, 25, 36, students see the number g the area of the squares. The number added to one (1), which is our (4), which is the number of squares needed to be added to r square. These numbers are also the square of 1,2, 3, 4, etc.

right triangles are place on top of the equilateral triangle, another formed. angle = wl, however, the area of a triangle = ½ wl because in a o triangles. Therefore, you must divide wl in ½ to find the area of

plane that do not intersect. h parallel and congruent opposite sides. rner on the right side of the parallelogram is cut and placed on the logram to form a square. This proves why we can use the same

ea of a square and a parallelogram.

.75 cm2

.25 52.25 x 3 =156.75 = 11 x 9.5 = 104.5 / 2 = 52.25 I = 22 x 9.5 = 209/2 = 104.5 = 104.5 + 52.25 = 156.75

multiplied by 2 = 156.75 x 2 = 313.5 le multiplied by 6 = ½ (11) (9.5) = 52.25 x 6 = 313.5

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Learning Experience 7 continued Page 4 21.

Length Width Area 5 2 10

6 3 18

3 2.5 7.5

7 6 42

x x+3 x (x+3)

y y+4 y (y + 4)

22. A = ½ (x+3) (x+2) or (x+3) ( x+2) or .5 (x+3) (x+2) 2 23. A = (x+3) + (x+1) x = 2x + 4 = (x+2)x 2 2

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Learning Experience 8: Modeling the Planets Objective: Students will use scientific notation to rewrite the diameter of the planets/distance of the planets from the sun and create cubic models of the planetary bodies and find the distance between them.

In this learning experience, students will create cubic models of the planetary bodies in our solar system. Students are given a chart on the activity sheet for Learning Experience #8 in the Cardstock Modeling Student Activity Book showing the diameter of the planetary bodies in kilometers. Students are then changing these very large numbers into scientific notation. To then create models of the planetary bodies, students are using the diameter of the planets and by moving the decimal showing them at 1/1000 their actual size (size in megameters). Then round the number to the nearest tenth to make the creation of the panels a little bit easier. After they round the number to the nearest tenth, that is the number they will use to create the panels. Students must remember that there is to be an 8 mm edge on the panel so the rubberbands can attach to the panel. Therefore, students can add 1.6 cm to the number they will use to create the panel to find the panels full size. For example, the panel for Mercury would be measured as shown below.

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Materials: For each group of three students: 3 Cardstock Modeling Student Activity Books Metric rulers Cardstock or posterboard (Neptune model) Glue* White paper* Pencil* Scissors*

For the class: 5 glue sticks 5 1/8” hole punch 5 ¼” hole punch 12 8 mm wooden sticks 12 binder clips Saw and Jig

*provided by teacher

Preparation: Demonstrate for students how to use the 8 mm wood sticks and saw/jig in creating the larger models of planetary bodies (Jupiter and Saturn).

Basic Skills Development: Manipulating Materials Measuring Comparing

Evaluation Strategy: Students will rewrite data of planetary body diameter in scientific notation and create cubic models of planetary bodies and the distances between them to scale.

Vocabulary: cube scientific notation diameter distance model template

Learning Experience 8 continued Page 2 Student groups are to be assigned a planet and/or the Moon to recreate. Due to their large size, Jupiter and Saturn should be created as a whole class project. The Sun, due to its large size, will not be recreated in this learning experience, however, taking the students out to the playground or gym to measure the diameter would give students true appreciation of its size. Students should first create a template of what each of their panels will look like. This can be done by making very exact measurements of their panels on white paper and gluing them on cardstock. Students are to share the 1/8” hole punch to punch smaller holes in the corners of the interior square of the panel. The six (6) panels for their cube can then be created by the group from this template. The panels are to be assembled into cubes with rubberbands to form the model of the planet/Moon their group has been assigned. The group that is creating the Neptune model will need to use poster board to create their panels and glue these sides along the 8 mm edge since the rubberbands are not long enough to hold the panels together. Small binder clips can be used along the side to hold the sides together until the glue dries. When Jupiter and Saturn are created by the class, students will discover that there is not cardstock large enough to create these panels. The class is to then create 16 additional card models of planet Earth to be used as corners for the large cube models of Jupiter and Saturn (8 cube models of Earth for Jupiter and 8 models of Earth for Saturn). We must use 8 mm wood sticks to create the length of each side of the Jupiter and Saturn models. The wood sticks are to be glued to the 8 mm edge of the panels of the Earth models in the corners of the models. To achieve the longer length, the wood sticks can be glued together and bound with two (2) cardboard pieces that are 8 mm long x 16 mm wide. These pieces can then be folded in half and one piece is to be glued on one side of the wood sticks and the second piece is to be glued on the opposite side at the location they are joined together.

Wood sticks

Models of Earth

A saw and jig are provided to cut wood sticks to appropriatare created, students will not only be able to get a sense ofbut they also will be able to make some comparisons betwemodel used as corners to whole size of Jupiter, Pluto fits inVenus are very close in size, etc.)

Cardboard piece to be glued where wood sticks are joined.

16 mm

8 mm fold

e lengths. Once the models the size of these planets, en them (compare Earth

to the Moon, Earth and

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Learning Experience 8 continued Page 3 In the final part of this learning experience, students are looking at the distance between the planets. Students are then moving the decimal to the place indicated with scientific notation on the chart. Then using the number that is 1/100,000,000 the actual size, students are to use maps to find out where these planetary bodies would be located from the sun and in relation to each other. Students then write down various things they have learned about the solar system from the model they created. It may be that they did not realize the size of some of the planets in relation to each other. It may be that they see the distance between the planets that are further from the sun are further away from each other than the planets closer to the sun, etc. Answers to activity sheet: 1.

Planetary Bodies Diameter of Planetary Bodies (km)

Diameter rounded to

nearest hundred Diameters written in scientific notation

Sun 1,391,000 1,391,000 1.391 x 106 Mercury 4878 4900 4.9 x 103 Venus 12,104 12,100 1.21 x 104 Earth 12,756 12,800 1.28 x 104 Mars 6,787 6,800 6.8 x 103

Jupiter 142,984 143,000 1.43 x 105 Saturn 120,660 120,700 1.207 x 105 Uranus 51,118 51,100 5.11 x 104

Neptune 49,528 49,500 4.95 x 104 Pluto 2,274 2,300 2.3 x 103 Moon 3,476 3,500 3.5 x 103

2.

Planetary Bodies

Diameter of Planetary Bodies (km)

1/1000 size (megameters)

Round reduced size to the nearest tenth.

Sun 1,391,000 1391 1391

Mercury 4878 4.878 4.9

Venus 12,104 12.104 12.1

Earth 12,756 12.756 12.8

Mars 6,787 6.787 6.8

Jupiter 142,984 142.984 143

Saturn 120,660 120.66 120.7

Uranus 51,118 51.118 51.1

Neptune 49,528 49.528 49.5

Pluto 2,274 2.274 2.3

Moon 3,476 3.476 3.5

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Learning Experience 8 continued Page 4 3.

Planetary Bodies

Distance from Sun (km) 103 105 108

Mercury 57910000 57910 579.1 .5791 Venus 108200000 108200 1082 1.082 Earth 149597871 149597.871 1495.97871 1.495978701 Mars 227940000 227940 2279.4 2.2794

Jupiter 778330000 778330 7783.3 7.7833 Saturn 1429400000 1429400 14294 14.294 Uranus 2870990000 28770990 28709.9 28.7099

Neptune 4504000000 4504000 45040 45.04 Pluto 5913520000 5913520 59135.2 59.132

4. Answers will vary. An example is if your school was the Sun, Mercury would be located .579 or approximately 6 km from your school. 5. Answers will vary. An example is Mercury = .579 km from the Sun; Venus = 1.08 km from the Sun. Subtract the two numbers to find out the distance they are from each other = .501. If your school was Mercury, find the distance that is .501 km from your school to show where Venus would be. 6. Answers will vary.

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GLOSSARY Area the measure of how much surface is covered by a figure. The number of square units a figure contains. This is often found by multiplying length and width. Base the lower side or face of geometric figure. Centimeter a metric unit of length; 100 centimeters equals one meter. Concentric having a common center, such as two circles, one inside the other. Congruent used to describe identical geometric shapes. They are the same in size and shape. Their sides are equal in length and their angles are equal in measure. Corner point where line segments meet. Cube a prism that has six square faces. Customary commonly practiced, used or observed. Diameter a straight line running from one side of rounded geometric

figure through the center to the other side. Difference the amount by which one quantity is greater or smaller than

another. Distance the length of space between two objects. Dodecahedron a three-dimensional geometric figure with 12 equal

pentagonal faces meeting in threes at 20 vertices. Edge the segment where two faces of a solid figure meet. Equilangular where all the angles in a shape have the same number of degrees of measure. Equilateral where all the sides of a shape are the same length. Equilateral triangle a triangle with three congruent sides, and three equal

angles. Exterior angle an angle on the outside of a polygon formed between a side

and an extension of an adjacent side.

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Face a flat surface of a three-dimensional figure. Form three-dimensional model of a shape. Hexagon a polygon with six sides. Hexahedron a three-dimensional geometric figure that has six plane

faces, for example, a cube. Icosahedron a three-dimensional geometric figure having 20 sides or

faces. Interior angle the angle formed between two adjacent sides of a polygon

and lying in its interior. The sum of the interior angles of any polygon is equal to the number of its sides minus two and multiplied by 180º.

Interior line segment the inside line segment in a diagram. Intersection where two lines meet or cross. Lateral pertaining to the side or sides. Line segment part of a line consisting of two endpoints and all the points

between them. Model a representation of an object made to a larger or smaller

scale than the original. Octahedron a 3-dimensional geometric figure that has eight faces. Panel flat part/side of three-dimensional forms. Parallel lines in the same plane that do not intersect. Parallelogram a four-sided plane figure in which both pairs of opposite

sides are parallel and of equal length and opposite angles are equal.

Pentagon a polygon that has five sides. Perimeter the linear distance around an object or figure. Also the boundary of a closed plane figure. Perpendicular two lines intersecting to form right angles.

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Platonic solids consist of the five regular polyhedrons: cube, tetrahedron, octahedron, icosahedron, and dodecahedron. Point an exact place or position in space represented by a dot. Polygon a closed figure formed from line segments that meet only at

their endpoints. Polygons form the faces of a polyhedron.

Polyhedron a three-dimensional geometric figure with many flat surfaces, formed by joining edges of polygons to enclose a region of space.

Prism a three-dimensional figure that has two congruent and parallel faces that are polygons. The remaining faces are parallelograms. Proportional having equivalent ratios. Pyramid a polyhedron whose base is a polygon and whose other

faces are triangles that share a common vertex. Quadrilateral two-dimensional geometric figure with four sides. Radius the segment or the length of the segment from the center of a circle to a point on the circle. Rectangle a polygon having four polygon sides and four right angles. Regular polygon a polygon in which all sides have the same length and all

angles have the same number of degrees of measure. Right angle an angle with a measure of 90 degrees. Scientific Notation short way of expressing large numbers. A way of expressing

a given number as a number between 1 and 10 multiplied by 10 to the appropriate power.

Square a polygon with four right angles and four equal sides. Surface area the total area of the faces (including bases) of a solid figure. Template a pattern or mold with one or more shapes used

to guide the manufacture or drawing of objects with a similar shape.

Tetrahedron a three-dimensional geometric figure that has four faces.

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Trapezoid a quadrilateral that has two parallel sides. Triangle a polygon with three sides. Vertex the corner point of an angle, polygon, or solid, where several edges meet. Vertices plural of vertex. Volume the amount of space occupied by a three-dimensional object

as measured in cubic units. This is often found by multiplying length, width and height.